On the rate of convergence in the global central limit theorem for random sums of independent random variables

We present upper bounds of the integral for 0 ae<currency> l ae<currency> 1 + delta, where 0 < delta ae<currency> 1, I broken vertical bar(x) is a standard normal distribution function, and Z (N) = is the normalized random sum with variance V S (N) > 0 (S (N) = X (1) + center dot center dot center dot + X (N) ) of centered independent random variables X (1) ,X (2) , . . . . The number of summands N is a nonnegative integer-valued random variable independent of X (1) ,X (2) , . . . .